TITLE: Upper bound on the l1 norm of a multivariate normal random variable
QUESTION [6 upvotes]: Let $X \sim {\cal N}_d(0, \sigma^2I_d)$. I am interested in bounding the tail probability $P[||X||_1 > t]$ from above. A pointer to a known exponential or polynomial tail bound would be appreciated.
One idea to deal with the above probability is to use something like McDiarmid's inequality, but suitable for unbounded random variables.

REPLY [9 votes]: A typical trick is to use Chebyshev's inequality with the function $\exp(\alpha x)$
to obtain 
$$P(\|X\|_1>t)\leq E(\exp(\alpha\|X\|_1))/\exp(\alpha t).$$
Since the 1-norm is a sum and the coordinates are independent, the right hand side equals 
$$\exp(-\alpha t)\ E(\exp(\alpha |Z|))^d=\exp(-\alpha t)\left(\exp(\alpha^2/2) (1+\mbox{erf}(\alpha/\sqrt{2})) \right)^d.$$
Here $Z$ is a one dimensional standard normal random variable. 
You can now try to optimize over $\alpha$. 
Substituting the cheap upper bound $\mbox{erf}(\alpha/\sqrt{2})\leq 1$, and then 
optimizing gives $\alpha=t/d$. Plugging this in we get 
$$ P(\|X\|_1>t)\leq 2^d \exp(-t^2/2d).$$
I know you could do better, but maybe it will suffice for your purposes.